#### Date of Submission

2-28-1973

#### Date of Award

2-28-1974

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Economic Research Unit (ERU-Kolkata)

#### Supervisor

Maitra, Ashok (ERU-Kolkata; ISI)

#### Abstract (Summary of the Work)

During the last fifteen years a large number of papers have been devoted to the study of elosed set valued multifunetions. The studies were notivated by both the theoretical and spplica- tional interests that such multifunetions have. From the appli- cational point of view it is worth noting that such multifune- tions arise in varlous problems of control theory, dynanie progra- aning etc. The theoretieal'aspects of these studies belong properly to classieal deseriptive set theory. Classionl deserip- tive set theory asks que stions about how sets are constructed and about other definability properties of sets. The results on closed set valued multifunctions asserting the existence of neasurable selectors or asserting that closed set valued multi- functions can be expressed as inages under CarathÃ©odory maps of suitable spaces, whose deseriptive nature is in sone sense simples vould fall in this category.It is known that most of the pleasant deseriptive proper- ties of elosed set valued multifunetions fail to hold for more general aultifunetions. Indeed there are many examples and some will be given in this thesis to show that P. valued sulti- functions do not have these pleasant properties. A certain munber of positive results are known about o-compact valued multifunetions.The se go back to lnte 30s and early 40s and can be found in thework of Kunugui, Novikov, Arsenin and Shehegolkov. Sone further positive rosuts abeut o-compact yalued nultifutetions are Ineluded in this thesis.The onea of 0, valued multifuhetions is essentially an unexplored territory and in this thesis we initiate a detailed study of the struc ture of such multifunetions.The main problams that are.considered in this thesis are (1) the existonee of a measurable selec tor for G valued multi- Tunetions, (11) the existence of neasurable cross sections for partitions into a, bats, (111) the ropresentation of G, valued multifunetions as images.under, special Carathdodory maps of particularly simple spaces, (1v) the Borel arame trization of , valued multifunctions, (v) the representation of, G. valued multifunetions as contimuous inages of closed set valued multi- Functions.The thesis is organised as follovs:In chapter o ve fix sone terninology and notation and state some known results. These are tsed throughout the thesis. Chapter 1 is tmainly concernod with the problom of represen- tation of closed valued multifunotions as inages under Carath- dodory naps of simple spaces. Here we are ablb to give various, refinements of a result of Ioffe [8]. One such theoren reads asLet T be a non-empty set, a fiela of subsets of ? and Xa Polish space. If P:T X is a closed valuods r:T x2 X -mensurable multifunction then there is a map such that for each te T, f(t..) is s continuous map fron E onto F(t) nnd for each O6 I, f(.so) is -me asurable, vhere * deno tes the countably ad1itive fanily of subsets of T generated by2 and 2 denotes the space of irrationsls.One of the 1nte resting corollaries to this result is the folloving:Let T be a netric space snd X a Polish space.

#### Control Number

ISILib-TH17

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Srivastava, Sheshi Mohan Dr., "Studies in the Theory of Measurable Maltifunctions." (1974). *Doctoral Theses*. 358.

https://digitalcommons.isical.ac.in/doctoral-theses/358

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843470